$\cos q = \frac{e^{i q} + e^{- i q}}{2}$
$\sin q = \frac{e^{i q} - e^{- i q}}{2 i}$
$e^{i π} = - 1$
$e^{i π / 2} = i$

$\cos x = - \sin x dx$
$\sin x = \cos x dx$

$d (a b) = a db + da b$

$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - ...$

1	128	1000 0000
2	192	1100 0000
3	224	1110 0000
4	240	1111 0000
5	248	1111 1000
6	252	1111 1100
7	254	1111 1110

$2^{1} = 2$ , $2^{2} = 4$ , $2^{3} = 8$ , $2^{4} = 16$ , $2^{5} = 32$ , $2^{6} = 64$ , $2^{7} = 128$ , $2^{8} = 256$ ,

$2^{9} = 512$ , $2^{10} = 1024$ , $2^{11} = 2048$ , $2^{12} = 4096$ , $2^{13} = 8192$ , $2^{14} = 16384$ , $2^{15} = 32768$ ,

$a^{\frac{1}{2}} + b^{\frac{1}{2}} = (a + b + 2 (a b)^{\frac{1}{2}})^{\frac{1}{2}}$
$a^{\frac{1}{2}} - b^{\frac{1}{2}} = (a + b - 2 (a b)^{\frac{1}{2}})^{\frac{1}{2}}$
$(n + k)^{\frac{1}{2}} + (n - k)^{\frac{1}{2}}$
$= 2^{\frac{1}{2}} (n + (n^{2} - k^{2})^{\frac{1}{2}})^{\frac{1}{2}}$

gcd $(a, b) = c$ is equivalent to $a k_{1} - b k_{2} = c$ where $(k_{1}, k_{2}) = 1$

$\int_{- \infty}^{\infty} \frac{1}{1 + x^{2}} = π$

$\sum_{i = 0}^{n} a^{i} = \frac{a^{n + 1} - 1}{a - 1}$

spherical coordinates  
   (x,y,z) = r sin(θ) cos(ω), r sin(θ) sin(ω), r cos(θ)
   θ measured from the z-axis in plane ⊥ xy plane
   ω measured form the x-axis in xy plane

              k£n                    w£n
   (a+bi)ⁿ  =  å ⁿ C_k (-1)^k/2a^n-kb^k  + i å ⁿC_w (-1)^(w-1)/2a^n-wb^w
              k=0,2,4,...            w=1,3,5,...


                            n
   (a+b)²ⁿ  =  C(2n,n)aⁿbⁿ + å C(2n,k) ( a^2n-kb^k + a^kb^2n-k )
                            k=0
           =  a²ⁿ+b²ⁿ + ab( C(2n,1)(a^2(n-1)+b^2(n-1)) + ab( C(2n,2)(a^2(n-2)+b^2(n-2)) + ... abC(2n,n)

Let C2 be defined as a general binomial coefficient, useful in binomial series and C2(3,5)=0.

                   r-1 
   C2(p,r)  =  1/r!*Õ p-k  
                   k=0

Binomial Theorem expressed as a sum of odd and even components.

   j=0 or 1,
   (a+b)^2w+j  =

 w
 å C2(2w+j,2k)a^2kb^2w+j-2k + C2(2w+j,1+2k)a^1+2kb^2w+j-1-2k
k=0 

            1 + nx + n(n-1)x² + n(n-1)(n-2)x³  + ...
   (1+x)ⁿ =           ——        ——
                      2!        3!
             ¥   
          =  å C2(n,k)x^k
            k=0

   cosh²t - sinh²t = 1 solves x² - y² = 1

   ¥          ¥ i 
  (å a_k)²  =  å å a_i-ka_k 
  k=0       i=0 k=0

Binomial Theorem
$(1 + x)^{n} = 1 + n \frac{x}{1!} + n (n - 1) \frac{x^{2}}{2!} + n (n - 1) (n - 2) \frac{x^{3}}{3!} + ...$

Shift and Scaling on Sums and Integrals

${\sum\int}_{i = a}^{b} f (i) = {\sum\int}_{i = a - w}^{b - w} f (i + w)$ ${\sum\int}_{i = a w}^{b w} f (i) = {\sum\int}_{i = a}^{b} f (i w)$

$\sum ‖ x_{i} y_{i} ‖$ $\prod \max (‖ x_{i} y_{i} ‖, 1)$

$[a > b] = [a \leq b]$

formulae

Shift and Scaling on Sums and Integrals